Mixed results here.
First was in my shoulder. I have a rotator cuff problem. First appeared about 25 years ago, got a shot and the pain went away for over 20 years, from just one shot! However when it did return shots didn't help. I went to physical therapy and got on a regular exercise regime designed by my physical therapist. It slowly helped and after a year the pain was mostly gone, I still do the exercises and it seems to keep things under control. So for shoulder I certainly recommend exercise.
Both of my thumbs have arthritis pain. Tried shots in both, did no good. Doc said no exercise would help, so I had surgery on the right, my worst one. That was almost a year ago and can't say I am 100% happy, but it is better than before. Now contemplating what to do about the left.
Then my knees, I have had knee pain for a long time, not debilitating but it has slowly gotten worse. Tried exercise and at first it helped, but only for a while. Tried shots a few times, they work well for a few weeks but the pain returns, you can just get the shots every 3 months so now, on Monday I am getting them replaced.
Long story is based on my experience the shots are worth a try, might help, might not. Also depending on the joint the right exercising can help.
Best of luck to you!